Integration, a cornerstone of calculus, is essentially the reverse of differentiation. It's a process used for finding the original function when its derivative is known. This section delves into the integration of functions in the form $(ax+b)^n$, where $n$ is a rational number, excluding -1. This covers the integration of constant multiples, sums, and differences of functions.

Understanding Integration

Integration as the Reverse of Differentiation:

This concept involves determining a function when its derivative is known, effectively reversing the process of differentiation.

Notation and Terminology:

• The integral sign, "∫", represents the operation of integration.
• The expression to be integrated, followed by "dx", indicates integration with respect to x.
• "dx" in integration corresponds to the "dx" in the derivative notation $\frac{dy}{dx}$.

Types of Integrals

• Indefinite Integrals: These are integrals without specific limits, including a constant of integration, $+c$, to account for the loss of the original function's vertical position during differentiation.
• Definite Integrals: Integrals with specified upper and lower limits, used for calculating exact values, such as areas under curves. They exclude the constant $+c$.

Integration Formulas

• Basic Integral Form:

$\int a x^n \, dx = \frac{a x^{n+1}}{n+1} + c, \quad \text{where } n \neq -1.$

• Integral of a Function in the Form $(ax+b)^n$:

$\int a x^n \, dx = \frac{a x^{n+1}}{n+1} + c$

again, $n \neq -1.$

Worked Examples

Example 1:

Find the equation of a curve $y$ in terms of $x$ that passes through the point (1, 3), given$\frac{dy}{dx} = 6x^2$.

Solution:

1. Integration:

$y = \int 6x^2 \, dx = \frac{6x^{3}}{3} + c = 2x^3 + c.$

2. Determining Constant $c$:

Using the point (1, 3):

$3 = 2(1)^3 + c.$

Solving for $c$:

$c = 1.$

3. Final Equation:

$y = 2x^3 + 1. $<p></p><h3><strong>Example 2:</strong> </h3><p>Calculate the integral of$5x^2 - 2x + 3$.</p><p></p><p><strong>Solution:</strong></p><p><strong>1. Integral Calculation</strong>: </p>$\int (5x^2 - 2x + 3) \, dx = \frac{5x^3}{3} - \frac{2x^2}{2} + 3x + c = \frac{5x^3}{3} - x^2 + 3x + c.$<p></p><p><strong>2. Simplified Equation: </strong></p>$\therefore \int (5x^2 - 2x + 3) \, dx = \frac{5x^3}{3} - x^2 + 3x + c.$